A Framework for Reasoning on ProbabilisticDescription Logics

Giuseppe Cota1[0000−0002−3780−6265], Riccardo Zese2[0000−0001−8352−6304], ElenaBellodi2[0000−0002−3717−3779], Evelina Lamma2[0000−0003−2747−4292], and

Fabrizio Riguzzi3[0000−0003−1654−9703]

1 Dipartimento di Scienze Matematiche, Fisiche e Informatiche – University of ParmaParco Area delle Scienze, 53/A, 43124 Parma

giuseppe.cota@unipr.it2 Dipartimento di Ingegneria – University of Ferrara

Via Saragat 1, I-44122, Ferrara, Italy3 Dipartimento di Matematica e Informatica – University of Ferrara

Via Saragat 1, I-44122, Ferrara, Italy

Abstract. While there exist several reasoners for Description Logics,very few of them can cope with uncertainty. BUNDLE is an inferenceframework that can exploit several OWL (non-probabilistic) reasonersto perform inference over Probabilistic Description Logics.In this chapter, we report the latest advances implemented in BUN-DLE. In particular, BUNDLE can now interface with the reasoners ofthe TRILL system, thus providing a uniform method to execute proba-bilistic queries using different settings. BUNDLE can be easily extendedand can be used either as a standalone desktop application or as a libraryin OWL API-based applications that need to reason over ProbabilisticDescription Logics.The reasoning performance heavily depends on the reasoner and methodused to compute the probability. We provide a comparison of the differentreasoning settings on several datasets.

Keywords: Probabilistic Description Logic, Semantic Web, Reasoner

1 Introduction

The aim of the Semantic Web is to make information available in a form that isunderstandable and automatically manageable by machines. In order to realizethis vision, the W3C has supported the development of a family of knowledgerepresentation formalisms of increasing complexity for defining ontologies, calledOWL (Web Ontology Languages), that are based on Description Logics (DLs).Many inference systems, generally called reasoners, have been proposed to reasonupon these ontologies, such as Pellet [33], Hermit [32] and Fact++ [34].

Nonetheless, modeling real-world domains requires dealing with informationthat is incomplete or that comes from sources with different trust levels. Thismotivates the need for the management of uncertainty in the Semantic Web,

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and many proposals have appeared for combining probability theory with OWLlanguages, or with the underlying DLs [24,17,21,10,23]. Among them, in [27,36]we introduced the DISPONTE semantics, which applies the distribution seman-tics [30] to DLs. Examples of systems that perform probabilistic logic inferenceunder DISPONTE are BUNDLE [27,28,36], and TRILL [38,36,37]. The formeris implemented in Java, the latter in Prolog.

To perform exact probabilistic inference over DISPONTE knowledge bases(KBs), it is necessary to perform either one of the following reasoning tasks:justification finding or pinpointing formula extraction. The former method con-sists of finding the covering set of justifications, i.e., the set of all justificationsfor the entailment of the query. In the latter, a monotonic boolean formula isbuilt, which compactly represents the covering set of justifications. Both of thesenon-standard reasoning tasks can be performed by a non-probabilistic OWL rea-soner. The first version of BUNDLE was able to execute justification finding byexploiting the Pellet reasoner only [33]. Then it was extended to exploit differentnon-probabilistic OWL reasoners and approaches for justification finding [7]. Inparticular, it embeds Pellet, Hermit, Fact++ and JFact as OWL reasoners, andthree justification generators, namely GlassBox (only for Pellet), BlackBox andOWL Explanation.

In this chapter, we illustrate the state of the art of the BUNDLE framework.In particular, we present a newer version of BUNDLE which also interfaces withthe probabilistic reasoners of the TRILL system, namely: (i) TRILL [38,36],which solves the justification finding problem, (ii) TRILLP [38,36], which re-turns the pinpointing formula using the approach defined in [2,3], and (iii) TOR-NADO [37], which, similarly to TRILLP, returns the pinpointing formula, butthe formula is represented in a way that can be directly used to compute theprobability. In this way, the user can run probabilistic queries in a uniform wayby using the preferred reasoner. In addition, BUNDLE can be easily extendedby “plugging-in” a new reasoner or by including new concrete implementationsof algorithms for justification finding/axiom pinpointing.

The performance of reasoning heavily depends on the reasoner and methodused to compute the probability for a given query and it is of foremost importancefor (distributed) probabilistic rule learning systems such as [29,8]. To evaluate thesystem, we performed several experiments on various real-world and syntheticdatasets using different settings.

The chapter is organized as follows: Section 2 briefly introduces DLs, whileSection 3 illustrates the problems of justification finding and pinpointing formulaextraction. Section 4 and Section 5 present DISPONTE and the theoretical as-pects of inference in DISPONTE KBs respectively. The description of BUNDLEis provided in Section 6. Finally, Section 7 shows the experimental evaluationand Section 8 concludes the paper.

A Framework for Reasoning on Probabilistic Description Logics 3

2 Description Logics

An ontology describes the concepts of the domain of interest and their rela-tions with a formalism that allows information to be processable by machines.The Web Ontology Language (OWL) is a knowledge representation language forauthoring ontologies or knowledge bases. The latest version of this language,OWL 2 [35], is a W3C recommendation since 2012. In order to reach some com-putational properties, it is possible to define a sublanguage of OWL 2, also calledprofile, by applying syntactic restrictions.

Descriptions Logics (DLs) provide a logical formalism for knowledge repre-sentation. They are useful in all the domains where it is necessary to representinformation and perform inference on it, such as software engineering, medicaldiagnosis, digital libraries, databases and Web-based informative systems. Theypossess nice computational properties such as decidability and (for some DLs)low complexity [4].

There are many different DL languages that differ in the constructs that areallowed for defining concepts (sets of individuals of the domain) and roles (sets ofpairs of individuals). The SROIQ(D) DL is one of the most common fragments;it was introduced by Horrocks et al. in [14] and it is of particular importancebecause it is semantically equivalent to OWL 2.

Let us consider a set of atomic concepts C, a set of atomic roles R and aset of individuals I. A role could be an atomic role R ∈ R, the inverse R− ofan atomic role R ∈ R or a complex role R ◦ S. We use R− to denote the set ofall inverses of roles in R. Each A ∈ A, ⊥ and > are concepts and if a ∈ I, then{a} is a concept called nominal. If C, C1 and C2 are concepts and R ∈ R∪R−,then (C1 uC2), (C1 tC2) and ¬C are concepts, as well as ∃R.C, ∀R.C, ∃R.Self,≥ nR.C and ≤ nR.C for an integer n ≥ 0.

A knowledge base (KB) K = (T ,R,A) consists of a TBox T , an RBox Rand an ABox A. An RBox R is a finite set of transitivity axioms Trans(R), roleasimmetricity axioms Asy(R), role disjointness axioms Dis(R,S), role inclusionaxioms R v S and role chain axioms R ◦ P v S, where R,P, S ∈ R ∪R−. ATBox T is a finite set of concept inclusion axioms C v D, where C and D areconcepts. An ABox A is a finite set of concept membership axioms a : C androle membership axioms (a, b) : R, where C is a concept, R ∈ R and a, b ∈ I.

A KB is usually assigned a semantics using interpretations of the form I =(∆I , ·I), where ∆I is a non-empty domain and ·I is the interpretation functionthat assigns an element in ∆I to each individual a, a subset of ∆I to eachconcept C and a subset of ∆I ×∆I to each role R. The mapping ·I is extendedto complex concepts as follows (where RI(x,C) = {y|〈x, y〉 ∈ RI , y ∈ CI} and

4 G. Cota, R. Zese et al.

#X denotes the cardinality of the set X):

>I = ∆I ⊥I = ∅{a}I = {aI} (¬C)I = ∆I \ CI

(C1 t C2)I = CI1 ∪ CI2 (C1 u C2)I = CI1 ∩ CI2(∃R.C)I = {x ∈ ∆I |RI(x) ∩ CI 6= ∅} (∀R.C)I = {x ∈ ∆I |RI(x) ⊆ CI}

(≥ nR.C)I = {x ∈ ∆I |#RI(x,C) ≥ n} (≤ nR.C)I = {x ∈ ∆I |#RI(x,C) ≤ n}(R−)I = {〈y, x〉|〈x, y〉 ∈ RI} (R1 ◦ ... ◦Rn)I = RI1 ◦ ... ◦RIn

(∃R.Self)I = {x|〈x, x〉 ∈ RI}

SROIQ(D) also permits the definition of datatype roles, which connect anindividual to an element of a datatype such as integers, floats, etc.

A query Q over a KB K is usually an axiom for which we want to test theentailment from the KB, written as K |= Q.

Example 1. Consider the following KB “Crime and Punishment” containing 4axioms αi

α1 = Nihilist v GreatMan α2 = ∃killed.> v Nihilist

α3 = (raskolnikov, alyona) : killed α4 = (raskolnikov, lizaveta) : killed

This KB states that if you killed someone then you are a nihilist and whoever isa nihilist is a “great man” (TBox). It also states that Raskolnikov killed Alyonaand Lizaveta (ABox). The KB entails the query Q = raskolnikov : GreatMan (butare we sure about that?).

2.1 Decidability of SROIQ

In order to ensure decidability of inferencing in SROIQ DL, three conditionsmust be met:

– no cardinality restrictions must be applied on transitive roles or on roles thathave transitive subroles [16,15];

– the RBox must be regular [14];– in the class expressions ≥ nR.C, ≤ nR.C and ∃R.Self, and in the axioms∃R.Self v ⊥ (also known as role irreflexivity axiom Irr(R)), Asy(R) andDis(R), R must be simple.

Definition 1 (Regular RBox [14]).

– An RBox is regular if there is a strict linear order ≺ on roles and the RBoxcontains only role chain axioms of the following forms:

R ◦R v R S− v RS1 ◦ S2 ◦ ... ◦ Sn v R R ◦ S1 ◦ S2 ◦ ... ◦ Sn v RS1 ◦ S2 ◦ ... ◦ Sn ◦R v R

where Si ≺ R for all i = 1, 2, . . . , n.

A Framework for Reasoning on Probabilistic Description Logics 5

Definition 2 (Simple role in SROIQ(D) [14]). Given a role R, its simplic-ity is inductively defined as follows:

– R is simple if it does not occur on the right hand side of a role inclusion axiomin R, i.e. there is no role chain axiom of the form: S1 ◦ S2 ◦ · · · ◦ Sn v R;

– an inverse role R− is simple if R is, and– if R occurs on the right hand side of a role inclusion axiom in R, then R is

simple if, for each S v R, S is a simple role.

3 Justification Finding and Pinpointing Formula

3.1 Justification Finding

Here we discuss the problem of finding the covering set of justifications for agiven query. This non-standard reasoning service is also known as axiom pin-pointing [31] and it is useful for tracing derivations and debugging ontologies.This problem has been investigated by various authors [18,31,5,13]. A justifica-tion corresponds to an explanation for a query Q. An explanation is a subsetof logical axioms E of a KB K such that E |= Q, whereas a justification is anexplanation such that it is minimal w.r.t. set inclusion. Formally, we say that anexplanation J ⊆ K is a justification if for all J ′ ⊂ J , J ′ 6|= Q, i.e. J ′ is not anexplanation for Q . The problem of enumerating all justifications that entail agiven query is called axiom pinpointing or justification finding. The set of all thejustifications for the query Q is the covering set of justifications for Q. Given aKB K, the covering set of justifications for Q is denoted by All-Just(Q,K).

Below, we provide the formal definitions of justification finding problem.

Definition 3 (Justification finding problem).Input: A knowledge base K, and an axiom Q such that K |= Q.Output: The set All-Just(Q,K) of all the justifications for Q in K.

Example 2. Consider the KB and the query Q = raskolnikov : GreatMan of Ex-ample 1. All-Just(Q,K) = { {α1, α2, α3}, {α1, α2, α4} }.

There are two categories of algorithms for finding a single justification: glass-box [18] and black-box algorithms. The former category is reasoner-dependent,i.e. a glass-box algorithm implementation depends on a specific reasoner, whereasa black-box algorithm is reasoner-independent, i.e. it can be used with any rea-soner. In both cases, we still need a reasoner to obtain a justification.

It is possible to incrementally compute all justifications for an entailment byusing Reiter’s Hitting Set Tree (HST) algorithm [26]. This algorithm repeatedlycalls a glass-box or a black-box algorithm which builds a new justification. Toavoid the extraction of already found justifications, at each iteration the extrac-tion process is performed on a KB from which some axioms are removed bytaking into account the previously found justifications. For instance, given a KBK and a query Q, if the justification J = {β1, β2, β3} was found, where βis areaxioms, to avoid the generation of the same justification, the HST algorithm

6 G. Cota, R. Zese et al.

tries to find a new justification on K′ = K r β1. If no new justification is foundthe HST algorithm backtracks and tries to find another justification by removingother axioms from J , one at a time.

3.2 Pinpointing Formula

Given a query, instead of finding the covering set of justifications, we can com-pute the pinpointing formula, which compactly represents the covering set ofjustifications, following the approaches proposed in [2,3].

To build a pinpointing formula, first we have to associate a unique proposi-tional variable to every axiom E of the KB K, indicated with var(E). Let var(K)be the set of all the propositional variables associated with axioms in K, then thepinpointing formula is a monotone Boolean formula built using some or all of thevariables in var(K) and the conjunction and disjunction connectives. A valuationν of a set of variables var(K) is the set of propositional variables that are true,i.e., ν ⊆ var(K). For a valuation ν ⊆ var(K), let Kν := {E ∈ K|var(E) ∈ ν}.

Definition 4 (Pinpointing formula). Given a query Q and a KB K, a mono-tone Boolean formula φ over var(K) is called a pinpointing formula for Q if forevery valuation ν ⊆ var(K) it holds that Kν |= Q iff ν satisfies φ.

It is worth noting that a pinpointing formula could be directly generatedfrom the covering set of justifications. However, this formula may not be themost compact.

Example 3. Consider the KB and the query Q = raskolnikov : GreatMan of Ex-ample 1. If we associate a Boolean variable Xi with each axiom αi, a pinpoint-ing formula could be X1 ∧ X2 ∧ (X3 ∨ X4). Another (non-compact) pinpoint-ing formula could be directly generated from the covering set of justifications:(X1 ∧X2 ∧X3) ∨ (X1 ∧X2 ∧X4).

4 Probabilistic Description Logics

DISPONTE [27,36] applies the distribution semantics [30] to Probabilistic De-scription Logic KBs.

In DISPONTE, a probabilistic knowledge base K is a set of certain axiomsand probabilistic axioms. Certain axioms take the form of regular DL axioms,whereas probabilistic axioms take the form p :: E, where p ∈ [0, 1] and E is a DLaxiom. The probability p can be interpreted as an epistemic probability, i.e., asthe degree of our belief in the truth of axiom E. Another interpretation is thatp represents the trustworthiness level of the data source for the axiom E.

DISPONTE associates independent Boolean random variables to the DLaxioms. The set of axioms that have the random variable assigned to 1 constitutesa world. The probability of a world w is computed by multiplying the probabilitypi for each probabilistic axiom βi included in the world by the probability 1− pifor each probabilistic axiom βi not included in the world.

Below, we provide some formal definitions for DISPONTE.

A Framework for Reasoning on Probabilistic Description Logics 7

Definition 5 (Atomic choice). An atomic choice is a couple (βi, k) where βiis the ith probabilistic axiom and k ∈ {0, 1}. The variable k indicates whether βiis chosen to be included in a world (k = 1) or not (k = 0).

Definition 6 (Composite choice). A composite choice κ is a consistent setof atomic choices, i.e., (βi, k) ∈ κ, (βi,m) ∈ κ implies k = m (only one decisionis taken for each axiom).

The probability of composite choice κ is: P (κ) =∏

(βi,1)∈κ pi∏

(βi,0)∈κ(1 − pi),where pi is the probability associated with axiom βi, because the random vari-ables associated with axioms are independent.

Definition 7 (Selection). A selection σ is a total composite choice, i.e., itcontains an atomic choice (βi, k) for every probabilistic axiom of the theory. Aselection σ identifies a theory wσ called a world: wσ = C∪{βi|(βi, 1) ∈ σ}, whereC is the set of certain axioms.

P (wσ) is a probability distribution over worlds. Let us indicate with W theset of all worlds. The probability of Q is [27]: P (Q) =

∑w∈W:w|=Q P (w), i.e. the

probability of the query is the sum of the probabilities of the worlds in whichthe query is true.

Example 4. Let us consider the knowledge base and the query Q = raskolnikov :GreatMan of Example 1 where some of the axioms are probabilistic:

β1 = 0.2 :: Nihilist v GreatMan α1 = ∃killed.> v Nihilist

β2 = 0.6 :: (raskolnikov, alyona) : killed β3 = 0.7 :: (raskolnikov, lizaveta) : killed

Whoever is a nihilist is a “great man” with probability 0.2 (β1) and Raskolnikovkilled Alyona and Lizaveta with probability 0.6 and 0.7 respectively (β2 and β3).Moreover there is a certain axiom (α1). The KB has eight worlds and Q is truein three of them, corresponding to the selections:

{ {(β1, 1), (β2, 1), (β3, 1)}, {(β1, 1), (β2, 1), (β3, 0)}, {(β1, 1), (β2, 0), (β3, 1)} }

The probability is P (Q) = 0.2·0.6·0.7+0.2·0.6·(1−0.7)+0.2·(1−0.6)·0.7 = 0.176.

5 Inference in Probabilistic Description Logics

It is often infeasible to find all the worlds where the query is true. To reducereasoning time, inference algorithms find, instead, explanations for the queryand then compute the probability of the query from them. Below we providethe definitions of DISPONTE explanations and justifications, which are tightlyintertwined with the previous definitions of explanation and justification for thenon-probabilistic case.

Definition 8 (DISPONTE Explanation). A composite choice φ identifies aset of worlds ωφ = {wσ|σ ∈ S, σ ⊇ φ}, where S is the set of all selections. Wesay that φ is an explanation for Q if Q is entailed by every world of ωφ.

8 G. Cota, R. Zese et al.

Definition 9 (DISPONTE Justification). We say that an explanation γ isa justification if, for all γ′ ⊂ γ, γ′ is not an explanation for Q.

The set of worlds ωΦ identified by the set of explanations Φ is covering Qif every world wσ ∈ W in which Q is entailed is such that wσ ∈ ωΦ. In otherwords, a covering set Φ identifies all the worlds in which Q succeeds.

Two composite choices κ1 and κ2 are incompatible if their union is incon-sistent. For example, κ1 = {(βi, 1)} and κ2 = {(βi, 0)} are incompatible. A setK of composite choices is pairwise incompatible if for all κ1 ∈ K, κ2 ∈ K,κ1 6= κ2 implies that κ1 and κ2 are incompatible. The probability of a pairwiseincompatible set of composite choices K is P (K) =

∑κ∈K P (κ).

Given a query Q and a covering set of pairwise incompatible explanations Φ,the probability of Q is [27]:

P (Q) =∑

wσ∈ωΦ

P (wσ) = P (ωΦ) = P (Φ) =∑φ∈Φ

P (φ) (1)

Unfortunately, in general, explanations (and hence justifications) are notpairwise incompatible, therefore the covering set of justifications cannot be di-rectly used to compute the probability. Even more so, the pinpointing formula,which compactly encodes the covering set of justifications, cannot be directlyused for probability computation. The problem of calculating the probability ofa query is therefore reduced to that of finding a covering set of justifications orthe pinpointing formula and then transforming it into a covering set of pairwiseincompatible explanations.

We can think of using non-probabilistic reasoners for justification findingor pinpointing formula extraction, then consider only the probabilistic axiomsand transform the covering set of justifications or the pinpointing formula intoa pairwise incompatible covering set of explanations from which it is easy tocompute the probability.

Example 5. Consider the KB and the query Q = raskolnikov : GreatMan ofExample 4. If we use justification finding algorithms by ignoring the proba-bilistic annotations, we find the following non-probabilistic justifications: J ={ {β1, α1, β2}, {β1, α1, β3} }. Then we can translate them into DISPONTE justi-fications: Γ = { {(β1, 1), (β2, 1)}, {(β1, 1), (β3, 1)} }. Note that Γ is not pairwiseincompatible, therefore we cannot directly use Equation (1). The solution to thisproblem will be shown in the following section.

6 BUNDLE

The reasoner BUNDLE [27,28] computes the probability of a query w.r.t. DIS-PONTE KBs by first computing all the justifications for the query, then con-verting them into a pairwise incompatible covering set of explanations by build-ing a Binary Decision Diagram (BDD). Finally, it computes the probability bytraversing the BDD using function Probability described in [19]. A BDD for

A Framework for Reasoning on Probabilistic Description Logics 9

a function of Boolean variables is a rooted graph that has one level for eachBoolean variable. A node n has two children corresponding respectively to the1 value and the 0 value of the variable associated with the level of n. Whendrawing BDDs, the 0-branch is distinguished from the 1-branch by drawing itwith a dashed line. The leaves store either 0 or 1.

Example 6 (Example 4 cont.). Let us consider the KB and the query of Exam-ple 4. If we associate random variables X1 with axiom β1, X2 with β2 and X3

with β3, the BDD representing the set of explanations is shown in Figure 1. Byapplying function Probability [19] to this BDD we get

Probability(n3) = 0.7 · 1 + 0.3 · 0 = 0.7

Probability(n2) = 0.6 · 1 + 0.4 · 0.7 = 0.88

Probability(n1) = 0.2 · 0.88 + 0.8 · 0 = 0.176

and therefore P (Q) = Probability(n1) = 0.176, which corresponds to theprobability given by DISPONTE.

X1 n1

X2 n2

X3 n3

1 0

Fig. 1. BDD representing the set of explanations for the query of Example 4.

BUNDLE uses implementations of the HST algorithm to incrementally ob-tain all the justifications from non-probabilistic reasoner. Moreover, in the latestversion, BUNDLE can exploit the reasoners provided by the TRILL framework,thus providing a simple and uniform way to execute probabilistic queries byusing the preferred method and reasoner.

Figure 2 shows the new architecture of BUNDLE. The main novelties are theinterfaces for the probabilistic reasoners of the TRILL system. These reasonersdirectly return the probability of the query, therefore, when using one of theTRILL reasoner interfaces, the Probability Computation module, which convertsa covering set of justifications into a BDD, is not used.

BUNDLE now supports: (1) four different OWL reasoners: Pellet 2.5.0, Her-mit 1.3.8.413 [32], Fact++ 1.6.5 [34], and JFact 4.0.44; (2) three probabilisticreasoners TRILL, TRILLP and TORNADO, which exploit Prolog’s backtrack-ing feature to obtain the justifications or the pinpointing formula; and (3) threedifferent strategies for finding a justification using a non probabilistic reasoner,which are:4 http://jfact.sourceforge.net/

10 G. Cota, R. Zese et al.

User Interface

BUNDLE App

Pellet Hermit Fact++ JFact

Probability Computation

BUNDLE Library

OWLExplanation

GlassBox(Pellet)

BlackBox(OWLAPI)

TRILL

TORNADO

TRILLP

TRILL System

Custom Hitting Set Tree

TRILLinterface

TRILLP

interface

TORNADOinterface

TRILL Interfaces

Fig. 2. Software architecture of BUNDLE.

GlassBox A glass-box approach which depends on Pellet. It is a modified ver-sion of the GlassBoxExplanation class contained in the Pellet Explanationlibrary.

BlackBox A black-box approach offered by the OWL API5 [12]. The OWL APIis a Java API for the creation and manipulation of OWL 2 ontologies.

OWL Explanation A library that is part of the OWL Explanation Work-bench [13]. The latter also contains a Protege plugin, underpinned by thelibrary, that allows Protege users to find justifications for entailments in theirOWL 2 ontologies.

All the supported non-probabilistic reasoners can be paired with the BlackBoxor the OWL Explanation methods, while only Pellet can exploit the GlassBoxmethod.

To find all justifications using the GlassBox and BlackBox approaches, weextended the HSTExplanationGenerator class of the OWL API, which providesan implementation of the HST algorithm, in order to support annotated axioms(DISPONTE axioms are OWL axioms annotated with a probability). OWL Ex-planation, instead, already contains an HST implementation and a black-boxapproach that supports annotated axioms.

Table 1 provides an overview of the reasoners and methods supported by theBUNDLE framework.

BUNDLE can be easily extended in three main ways:

– By adding a non-probabilistic reasoner that implements the OWLReasoner

interface of the OWL API library.– By adding a probabilistic reasoner that implements the interface

ProbabilisticReasoner defined in BUNDLE. Indeed, the interfaces for theTRILL reasoners implement this Java interface.

5 http://owlcs.github.io/owlapi/

A Framework for Reasoning on Probabilistic Description Logics 11

Table 1. Reasoners supported by the BUNDLE framework. The symbol Xmeans thatthe reasoner is compatible with the method used for computing the probability.

Justification Finding Pinpointing Formula

Hitting Set Tree Prolog backtracking

Reasoner DL GlassBox BlackBox OWL Expl. built-in built-in

Pellet SROIQ(D) X X XHermit SROIQ(D) X XFact++ SROIQ(D) X XJFact SROIQ(D) X XTRILL SHIQ XTRILLP SHI XTORNADO SHI X

– By adding a non-probabilistic reasoner that is able to perform the rea-soning task of justification finding. This reasoner should implement theExplanationReasoner interface defined in BUNDLE.

BUNDLE can be used as standalone desktop application or as a library. More-over, we developed a web application available at http://bundle.ml.unife.it/in a similar way to what has been done for TRILL [6] and cplint [1]. The webapplication allows the user to test BUNDLE without installing any software onthe local machine.

6.1 Using BUNDLE as an Application

BUNDLE is an open-source software and is available on Bitbucket, together withits manual, at https://bitbucket.org/machinelearningunife/bundle.

A BUNDLE image was deployed in Docker Hub. Users can start using BUN-DLE by executing the following commands:

sudo docker pull giuseta/bundle :4.0.0

sudo docker run -it giuseta/bundle :4.0.0 bash

A bash shell of the container then starts and users can execute probabilisticqueries by running the command bundle.

6.2 Using BUNDLE as a Library

BUNDLE can also be used as a library. Once the developer has added BUN-DLE dependency in the project’s POM file, the probability of the query can beobtained in just few lines:

1 BundleConfigurationBuilder configBuilder = newBundleConfigurationBuilder(ontology);

2 BundleConfiguration config = configBuilder3 .hstMethod(hstMethod).reasoner(reasonerName)4 .buildConfiguration ();5 Bundle reasoner = new Bundle(config);6 reasoner.init();7 QueryResult result = reasoner.computeQuery(query);

12 G. Cota, R. Zese et al.

where ontology and query are objects of the classes OWLOntology andOWLAxiom of the OWL API library respectively.

Lines 1-4 show that the developer can inject the preferred HST method forjustification finding (none for the TRILL reasoners) and the favorite reasonerby using a configuration builder. After initialization (lines 5-6), probabilisticinference is performed (line 7).

7 Experiments

We performed four different tests to compare the possible configurations of BUN-DLE, which depend on the reasoner and the justification search strategy chosen,for a total of 12 combinations. For each query we set a timeout of 10 minutes. Inthe first test we compared all configurations on four different datasets, in order tohighlight which combination reasoner/strategy shows the best behavior in termsof inference time. To investigate the scalability of the different configurations, inthe last three experiments, we considered KBs of increasing size in terms of thenumber of probabilistic axioms.

All tests were performed on the HPC System Galileo6 equipped with IntelXeon E5-2697 v4 (Broadwell) @ 2.30 GHz, using 1 core for each test.

Test 1 The first test considers 4 real world KBs of various complexity as in [38]:(1) BRCA [20], which models the risk factors of breast cancer; (2) an extract ofDBPedia [22], containing structured information from Wikipedia, usually thosecontained in the information box on the righthand side of pages; (3) Biopaxlevel 3 [9], which models metabolic pathways; (4) Vicodi [25], which containsinformation on European history and models historical events and importantpersonalities.

We used a version of the DBPedia, Biopax and BRCA KBs without the ABoxand a version of Vicodi with an ABox of 19 individuals. For each KB we addeda probability annotation to each axiom. The probabilistic values of a KB wererandomly assigned using a uniform distribution U(0, 1) and are fixed for all thequeries performed on that KB. We randomly created 50 different subclass-ofqueries for BRCA, DBPedia and BioPax, and 50 different instance-of queries forVicodi, following the concepts hierarchy of the KBs, ensuring each query had atleast one explanation.

Table 2 shows the average time in seconds to answer queries with differentBUNDLE configurations. Bold values highlight the fastest configuration for eachKB. Cells with “–” indicate that the timeout was reached in at least one query.Cells with “crash”, instead, indicate that the reasoner experienced an internalerror and was not able to return a result.

Overall, the best results are obtained by Pellet with the GlassBox approach.However, the use of OWL Explanation library shows competitive results. ForBioPax and Vicodi KBs, the Fact++/BlackBox configuration was not able to

6 http://www.hpc.cineca.it/hardware/galileo

A Framework for Reasoning on Probabilistic Description Logics 13

return a result for any query. TRILL and TRILLP reached the timeout at leastin one query for BioPax and BRCA, while using TORNADO a timeout occurredwhen querying BioPax.

Table 2. Average time (in seconds) for probabilistic inference with all possible con-figurations of BUNDLE over different datasets (Test 1). “–” means that the executiontimed out (600 s). crash means that the reasoner experienced an internal error.

DatasetReasoner Method BioPax BRCA DBPedia Vicodi Average

Pellet GlassBox 1.316 0.886 0.623 0.956 0.945Pellet BlackBox 1.619 1.653 0.570 1.614 1.364Pellet OWLExp 1.070 1.034 0.914 1.420 1.110Hermit BlackBox 4.199 6.694 1.299 4.832 4.256Hermit OWLExp 1.198 2.071 0.835 2.024 1.532JFact BlackBox 1.648 1.852 0.541 1.341 1.346JFact OWLExp 0.887 1.012 0.878 1.179 0.989Fact++ BlackBox crash 0.649 0.363 crash n.a.Fact++ OWLExp 0.903 0.554 0.588 3.285 1.333TRILL – – 0.442 0.470 n.a.TRILLP – – 0.335 0.424 n.a.TORNADO – 2.439 0.287 0.373 n.a.

Test 2 The second test was performed following the approach presented in [20]on the BRCA KB (ALCHF(D), 490 axioms). To test BUNDLE, we randomlygenerated and added an increasing number of subclass-of probabilistic axioms.The number of these axioms was varied from 9 to 16, and, for each number,30 different consistent KBs were created. Every time a KB is generated, theprobability values of the axioms are generated anew. The number of additionalaxioms may cause an exponential increase of the inference complexity (please see[20] for a detailed explanation). In these tests we consider those possible caseswhere most of our knowledge is certain but there are some uncertainties.

Finally, an individual was added to every KB, randomly assigned to eachsimple class that appeared in the probabilistic axioms, and a random probabilitywas attached to it. We ran 60 probabilistic queries of the form a : C where a is theadded individual and C is a class randomly selected among those that representwomen under increased and lifetime risk such as WomanUnderLifetimeBRCRiskand WomanUnderStronglyIncreasedBRCRisk, which are at the top of the concepthierarchy.

Table 3 shows the execution time averaged over the 60 queries as a functionof the number of probabilistic axioms. For each size, bold values indicate thebest configuration. The results show that TORNADO is the only reasoner thatcan provide answers by respecting the time limits and all its execution timingswhere competitive with the best configurations.

14 G. Cota, R. Zese et al.

Table 3. Average execution time (in seconds) for probabilistic inference with differentconfigurations of BUNDLE on versions of the BRCA KB of increasing size (Test 2).“–” means that the execution timed out (600 s).

Reasoner Method 9 10 11 12 13 14 15 16

Pellet GlassBox 2.437 1.812 14.914 – 4.101 – 3.743 7.102Pellet BlackBox 8.082 5.152 29.151 – 14.470 – 12.453 18.727Pellet OWLExp 3.034 2.339 10.507 – 4.759 – 4.025 5.405Hermit BlackBox 38.431 22.625 – – 73.854 – – –Hermit OWLExp 11.249 8.497 29.536 – 18.850 – – 19.692JFact BlackBox 8.937 5.759 35.504 – 16.451 – 15.096 23.108JFact OWLExp 3.068 2.384 8.494 – 4.493 – 3.978 5.103Fact++ BlackBox – – 17.015 – – – – –Fact++ OWLExp – – – – – – – 4.510TRLL – – – – – – – –TRILLP – – – – – – – –TORNADO 3.425 2.659 4.705 4.252 4.604 4.277 4.022 4.745

Test 3 In the third test we artificially created a set of KBs of increasing size ofthe following form:

(β1,i) 0.6 :: Bi−1 v Pi uQi (β2,i) 0.6 :: Pi v Bi (β3,i) 0.6 :: Qi v Bi

where n ≥ 1 and 1 ≤ i ≤ n. The query Q = B0 v Bn has 2n justifications, evenif the KB has a size that is linear in n. We increased n from 2 to 10 in steps of 2and we collected the running time, averaged over 50 executions. Table 4 shows,for each n, the average time in seconds that the systems took for computing theprobability of the query Q (in bold the best time for each size). Cells with “–”indicate that the timeout occurred at least in one query.

The experimental results show that using TORNADO outperforms the othersettings. If the size of the synthetic dataset is greater or equal to 10, all the runsreach a timeout.

Table 4. Average execution time (in seconds) for probabilistic inference with differentconfigurations of BUNDLE on synthetic datasets (Test 3). “–” means that the executiontimed out (600 s).

Reasoner Method 2 4 6 8 10

Pellet GlassBox 0.890 1.573 7.720 – –Pellet BlackBox 1.060 2.446 12.738 – –Pellet OWLExp 1.843 4.055 10.443 31.118 –Hermit BlackBox 5.968 29.316 168.286 – –Hermit OWLExp 4.410 16.637 63.062 239.256 –JFact BlackBox 1.039 2.205 10.978 – –JFact OWLExp 1.749 3.869 9.833 28.170 –Fact++ BlackBox – 2.229 – – –Fact++ OWLExp 1.728 – 10.159 30.923 –TRILL 0.549 1.244 3.708 34.105 –TRILLP 0.267 0.443 23.767 – –TORNADO 0.194 0.203 0.240 0.343 –

A Framework for Reasoning on Probabilistic Description Logics 15

Test 4 To further test the various settings of the BUNDLE framework on realworld KBs, we have conducted a test using the Foundational Model of AnatomyOntology (FMA for short)7. FMA is a KB for biomedical informatics that modelsthe phenotypic structure of the human body anatomy. It contains 4,706 axioms inthe TBox and RBox, with 2,626 different classes. To perform this test, we created7 versions of the KB containing an increasing number of individuals. The first5 versions contains a number of individuals varying from 20 to 100 with stepsof 20, while the last 2 versions contain 200 and 300 individuals. Each individualcan be the subject of 1 to 11 assertions. Then we added a probability annotationto each axiom with random values sampled from a uniform distribution U(0, 1).For each KB of a given size, we ran 10 times the query i0 : Organ zone, wherei0 is an individual which is present in all the KBs. The averaged running timeis reported in Table 5.

The results show that Pellet with GlassBox obtains the best performancesand that with OWL Explanation paired with any reasoner we obtain competitiveresults. However, it seems that the Prolog-based approaches have some issueshandling high numbers of individuals.

Table 5. Average execution time (in seconds) for probabilistic inference with differentconfigurations of BUNDLE on versions of the FMA KB of increasing size (Test 4). “–”means that the execution timed out (600 s).

Reasoner Method 20 40 60 80 100 200 300

Pellet GlassBox 1.392 1.491 1.611 1.594 2.457 1.890 1.931Pellet BlackBox 9.253 9.458 11.772 15.509 43.435 20.649 22.118Pellet OWLExp 2.000 2.004 2.091 2.092 5.757 2.371 2.470Hermit BlackBox 23.660 24.306 26.570 32.344 97.917 40.773 40.969Hermit OWLExp 3.913 3.907 3.950 4.201 10.349 4.169 4.184JFact BlackBox 7.715 8.206 9.612 12.129 31.382 17.708 17.547JFact OWLExp 1.900 1.985 1.956 2.003 4.995 2.290 2.294Fact++ BlackBox 8.219 – 9.526 11.676 34.216 16.953 –Fact++ OWLExp 1.644 1.687 1.689 1.774 4.648 1.853 1.845TRILL 17.286 23.912 35.932 59.228 93.456 – –TRILLP 17.884 31.431 50.811 89.426 142.585 – –TORNADO 17.035 24.192 37.112 62.409 97.258 – –

8 Conclusions

In this chapter, we illustrated the state of the art of BUNDLE, a framework forreasoning on Probabilistic Description Logics KBs that follow DISPONTE. Theframework can be used both as a standalone application and as a library andit allows to pair 4 different OWL reasoners with 3 different approaches to findquery justifications. Moreover, the latest version add the possibility of using theProlog-based probabilistic reasoners TRILL, TRILLP and TORNADO. We also

7 http://si.washington.edu/projects/fma

16 G. Cota, R. Zese et al.

developed a web application that allows to test BUNDLE without installing anysoftware on the local machine.

We provided a comparison between the various configurations reasoner/ap-proach over different datasets, showing that in 2 out of 4 experiments (Test 1 andTest 4) Pellet paired with GlassBox or any reasoner paired with the OWLExpla-nation library achieve the best results in terms of inference time on a probabilisticontology. However, for BRCA datasets of increasing size (Test 2) and for syn-thetic datasets with queries that have an exponential number of justifications(Test 3), TORNADO showed the best performance.

In the future, we plan to study the effects of glass-box or grey-box methods forcollecting explanations. Moreover, we plan to integrate in BUNDLE a reasonerbased on Abductive Logic Programming [11].

Acknowledgement This work was supported by the “GNCS-INdAM”.

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